Characterizing the Generalization Error of Random Feature Regression with Arbitrary Data-Augmentation

Data-driven modeling in real-world regression tasks often suffers from limited training samples, high collection costs, and noisy observations. Inspired by the impact of data augmentation in vision and language, we propose a novel Counterfactual Residual Data Augmentation (CRDA) technique for tabular regression. Our key insight is that once a regressor has modeled the systematic component of the data, the remaining noise can be viewed as an invariant residual that remains stable under small perturbations of carefully selected features. We exploit this residual invariance to generate new, yet realistic, training samples, effectively expanding the dataset without requiring additional real data. Our method is model-agnostic and readily applicable to various types of regressors. In experiments across datasets from a variety of benchmark repositories, on average, CRDA reduces an MLP Regressor's MSE by 22.9% and an XGBoost Regressor's MSE by 6.4%. When compared to existing state-of-the-art data generators and augmentation techniques, CRDA consistently outperforms in MSE reduction. By adding principled counterfactual variations to the training data, our method offers a simple and efficient remedy for noise-prone, small-sample regression settings.

Data augmentation is a simple and model-agnostic approach for exploiting known invariances in learning problems. Given a group acting on the input space, one augments the training set with transformed copies of each sample. Because it exploits symmetries without modifying the underlying learning algorithm, data augmentation can be applied broadly across learning methods. However, this universality comes at a computational cost: when the group is large, full group-sized augmentation quickly becomes computationally infeasible. This raises a fundamental question: Can partial data augmentation achieve the same statistical benefits as full augmentation in terms of generalization and sample complexity? We develop a general framework for investigating this question using Fourier analysis and the representation theory of finite groups. We show that, for a broad class of classical learning problems, partial data augmentation based on a randomly sampled subset of group elements achieves the same minimax rates as full augmentation, up to an approximation error that vanishes as the subset size increases. Our results provide a theoretical explanation for why partial augmentation can retain the statistical benefits of full augmentation despite enforcing symmetry only approximately, and shed light on a recently raised question in learning with symmetries: whether statistically optimal learning under general group invariances can be achieved using computationally scalable methods. Moreover, we prove a complementary impossibility result: enforcing exact invariance via data augmentation requires averaging over the entire group, and cannot be achieved by any strict subset when the hypothesis space is sufficiently expressive. Together, these results provide a unified perspective on full and partial data augmentation, as well as exact and approximate symmetry enforcement.

Self-supervised learning matches supervised accuracy from a fraction of the labels, but the labeled-sample efficiency behind this has lacked a theoretical explanation. We provide one. Data augmentation induces a similarity graph on the unlabeled data, so downstream learning on that graph is graph-Laplacian-regularized learning. We prove a fast transductive rate, $O(1/n_L)$ in the number of labels, in place of the supervised $O(1/\sqrt{n_L})$, by carrying the leave-one-out stability apparatus of Johnson and Zhang (JMLR 2007) over to the augmentation graph, and without the unrealistic assumptions of limit-based analyses (exact kernel, generalizing features). The bound makes augmentation quality explicit: the expected error is at most $C/n_L + R_{\mathrm{DA}}(y)$, where the data-augmentation alignment error $R_{\mathrm{DA}}(y)$ is proportional to the graph-cut mass of augmentations that cross a label boundary, so good augmentations let few labels suffice. The analysis uses a streamlined loss that drops the projector, negative-sample, and orthogonality overhead of standard objectives yet still recovers the top-$K$ ideal features in the infinite-data limit, the augmentation-kernel eigenspace studied by Zhai et al. The bound gives a mechanistic account of the accuracy-versus-label-count curve through augmentation quality, verified in a controlled model where the constants are known.